Problem: Solve for $x$ : $ 4|x - 10| + 4 = 3|x - 10| + 7 $
Solution: Subtract $ {3|x - 10|} $ from both sides: $ \begin{eqnarray} 4|x - 10| + 4 &=& 3|x - 10| + 7 \\ \\ { - 3|x - 10|} && { - 3|x - 10|} \\ \\ 1|x - 10| + 4 &=& 7 \end{eqnarray} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} 1|x - 10| + 4 &=& 7 \\ \\ { - 4} &=& { - 4} \\ \\ 1|x - 10| &=& 3 \end{eqnarray} $ Simplify: $ |x - 10| = 3$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -3 $ or $ x - 10 = 3 $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -3 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -3 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -3 + 10 \end{eqnarray} $ $ x = 7 $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = 3 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& 3 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& 3 + 10 \end{eqnarray} $ $ x = 13 $ Thus, the correct answer is $x = 7 $ or $x = 13 $.